Diversification measurement and analysis system

ABSTRACT

This disclosure details methods for measuring and analyzing diversification of portfolio of assets. A dimension is a logical and quantitative means to measure diversification. As the number of dimensions increases so does diversification. Strong asset correlations among each other detract from the notion of independence. A positive correlation increases risks and is therefore undesirable. Assets are embedded into a high dimensional Euclidean vector space. The entire portfolio is interpreted as a set of points whose ambient dimension is the number of assets in the portfolio. The Karhunen-Loève expansion is used to quantify the KL dimension of the geometric object induced by a portfolio. The associated dimension is taken as the measure of diversification accounts for both the number of assets and the commonality within them. This ensures that measuring diversification as a dimension accounts for the complete diversification affect of the portfolio and is thus a valuable portfolio management tool.

PRIORITY

This application is a continuation of, and claims priority from, U.S.Non-Provisional patent application Ser. No. 12/417,713, entitled“Diversification Measurement and Analysis System,” filed Apr. 3, 2009,which itself claimed priority from U.S. Provisional Patent ApplicationSer. No. 61/042,027, entitled “Diversification Measurement and AnalysisSystem,” filed Apr. 3, 2008. The disclosures of both of thoseapplications are hereby incorporated by reference in their entireties.

FIELD OF INVENTION

Embodiments of the invention concern software implemented systems,apparatuses, and methods in the field of investments and statistics.

BACKGROUND

Diversification is measured for portfolios of assets. Portfolios aremodeled geometrically and the dimension of the induced geometry is takenas the diversification measurement.

The measurement of total diversification has been absent from the realmof investments and statistics. Diversification is thus left as aninconsistent and qualitatively applied analysis technique and is usedimplicitly in traditional optimization techniques as a risk mitigatingcontrol.

Diversification is a powerful tool that reduces the variance of aportfolio and consequently helps to stabilize performance, potentiallyenabling more consistent returns and mitigating risk. Diversificationhas also been shown to be a cornerstone of judging the prudence of afiduciary.

Diversification is an investment attribute that generates greatconsensus as to its efficacy as a management attribute. Therefore, theutility of a robust and consistent measurement of diversification wouldbe strong. Despite the widespread acceptance of diversification,investors suffer from the absence of a well-defined, uniformquantitative metric. In fact, investors are accustomed to thinking ofdiversification as only an abstract or qualitative attribute.

A statistical measurement of the relationship of assets is an indicationof diversification. Yet, these relationships, traditionally co-variancesor correlations, measure a unique relationship between any two singleassets in the portfolio. Because the portfolio represents the entirecomposition of all these relationships, the measure of any one singlerelationship fails to represent a genuine level of portfoliodiversification.

Another common misinterpretation of diversification is Beta. Beta is ameasurement given to a portfolio that describes the amount ofperformance of that portfolio that can be explained by the marketforces. While Beta can be construed as a measurement of diversification,it too has some limitations. Beta requires the presence of anotherportfolio external to that portfolio being measured. Conventionally,this portfolio may be the S&P 500 or other broad index. Beta thus hassome utility for measuring the degree to which an asset will move withthe market. This index is thus an approximation of the market.

Beta has several problems, in that defining the market is inherentlyproblematic. In the United States alone, there are more than 200,000investment products. No investment index can even come close toaccounting for all of the different investment possibilities.Ultimately, every new IPO, every new start-up business and every newidea defines that market. No relative measure can ever capture the truemarket.

A further issue is that there is little efficacy in comparing theportfolio to the market. Beta is a relative measurement, whereas ameasurement of diversification that would better help investors toconstruct and manage portfolios for performance purposes would not beconcerned with items external to the portfolio but only those assetscomprising the portfolio. Therefore, a holistic measurement ofdiversification that was independent of any market benchmark or indexwould be desirable by investors seeking better performance.

Investors care more about what happens to their portfolio than what themarket does. Investors who have a sole focus on absolute returns areendeavoring to maximize the value of their portfolio. Such investorswould prefer a holistic measurement of diversification than a measurethat is relative to another (and essentially arbitrary) index.

Diversification has been primarily measured as to the number of assetsheld and to a lesser extent the largest allocations among those assets.Measuring diversification in this way fails to account for disparateweightings of assets and fails to account for the commonality of assets.To illustrate, consider a 10-asset portfolio consisting of ten equallyweighted portfolios, having each asset perform identically to anotherasset. This portfolio has the overall performance of only one asset,despite holding ten different investments.

Traditional and commonly used statistics can provide some insight intodiversification, but they fail to measure it. For example, measures ofdispersion, central tendency and distribution are in one sense measuresof diversification. Perhaps the most applicable among these measurementsis the kurtosis. The kurtosis is a value reflecting how close observedvalues are to the mean of those values. The kurtosis is the fourthmoment of a probability distribution and is therefore a dimensionlessquantity. This measure also fails as a robust measurement ofdiversification in that it cannot account for the full dimensionality ofthe underlying data. In the conventional application, a portfoliomanager would use the distribution and kurtosis to view the portfolioover time, and thus it does not provide any holistic insight.

Cluster analysis has some useful applications to help analyzediversification. However, cluster analysis fails to distill portfoliodiversification to a singular value that may then be used to aid in therelative analysis of portfolios. Distilling portfolio diversification toone singular value is thus desirable to aid in investment analysis andselection, optimization, attribution and presentation of portfolios andassets.

Investors currently have no way of measuring diversification except forthe Concentration co-Efficient (CC), Intra-Portfolio Correlation (IPC),and their derivates. A consistent, robust and quantitativediversification metric is thus of great utility to the industry.

Concentration Coefficient (CC)

The concentration coefficient (CC) measures portfolio concentration interms of the asset weightings. In an equal weighted portfolio, the CCwill be equal to the number of assets. As the portfolio becomes moreconcentrated in particular assets the CC will be proportionally reduced.

Thus, the Concentration Coefficient (CC) is defined as:

${CC}_{t}^{P} \equiv \left( {\sum\limits_{i = 1}^{N}\; \left( w_{i,t}^{P} \right)^{2}} \right)^{- 1}$

P is the portfolio

N is the number of stocks held in the portfolio

Wi,t is the weight of the i^(th) stock in the portfolio at time t

The concentration coefficient has the desirable property of being adiscrete measurable quantity. However, it fails to account for therelationships of assets and thus is inadequate for managingdiversification against market risk or systemic risk, which are amongthe most prevailing risks investors face, and need to be managed.

Intra-Portfolio Correlation (IPC)

Intra-portfolio correlation (IPC) is a means to quantifydiversification.

The range is from −1 to 1, with values approaching 1 being the leastdiversified. The IPC is a weighted average intra-portfolio correlation.

The Intra-Portfolio Correlation (IPC) statistic is calculated asfollows:

${IPC} = {\sum\limits_{i}\; {\sum\limits_{j}\; {X_{i}X_{j}p_{ij}}}}$

-   -   X_(i) is the fraction invested in asset i    -   X_(j) is the fraction invested in asset j    -   P_(ij) is the correlation between assets i and j    -   The expression may be computed when i≠j

The IPC is thus a measure of diversification against risks such assystemic risk but fails to account for other risks such as securityrisk, concentration risk and model risk.

Other Forms

Other forms of diversification analysis pertain to the classification ofan asset such as depicting the sector or asset class assigned toportfolio assets. Classification schemas may be economic sectors,industries, valuation models or geographic locations. In addition, andespecially within corporate portfolio models, elements may be productlines, the difference between these product lines in terms of elementssuch as price, characteristics of the targeted market, manufacturingstyle, goods or services, and materials used. Corporations also seekdiversification for their investor base, supplier base, employee base,customer base. All such analysis techniques are qualitative, notquantitative.

While useful, this analysis fails to entirely account fordiversification by subjugating analysis to only the studied elements,and again failing to provide a single measurement.

These techniques have additional drawbacks. They require categorizationof all components. Frequently, the categorization of such assets lacksrigor and suffers from a non-optimal division. Additionally, there is noconsistent process or categorization for determining asset classes,which portends a lack of consistency when describing the diversificationattributes of various portfolios. Therefore characterizingdiversification as having exposure to various asset classes results ininconsistent, non-comparable, and varying solutions.

A further element of subjectivity often results when mapping investmentsto categories. The inconsistent mapping process of various institutionsmap results in dissimilar diversification analyses.

These techniques fail to deliver one simple numerical value that aninvestor can use to analyze and compare portfolios. Therefore the resultof the analysis is inadequate to compare statistically similarportfolios as the investor would have to analyze an array of values andmost humans are incapable of accurately determining an optimal or evensuperior result, wherein if the result is reduced to one value, itscomprehension and comparative efficacy increase dramatically.

Types of Diversification

For business strategy purposes, diversification is sometimes categorizedas vertical, horizontal or concentric. In these conventional meanings,horizontal diversification is meant to mean broadening of the productline. Vertical diversification is the integration of the supply chain ordistribution outlets and concentric diversification is a corporategrowth strategy whereby a business builds its total sales by acquiringor establishing other unrelated businesses that may share management ortechnical efficiencies.

When the decision maker has a presumption of control, a diversificationstrategy can have several disadvantages. Namely as a corporationdiversifies, fewer resources are able to be devoted to the same assets.This can diminish the ability for any one asset to reach a criticalmass. A more microeconomic view may be that an asset with certain fixedand sunk costs may have a decreased or negative net present value whenother assets supporting the continued development and sales of thatasset are diluted by the new diversified strategy. One could imaginethat certain efficiencies and advantages can be obtained by practicingthe discipline of focus.

While the efficacy of maximizing diversification is not clear incorporate models, the utility of measuring diversification is clear.Accounting for diversification germane to a company provides the companywith valuable information that when combined with business intelligencecan shape strategy and execution. Product lines, target customers andbusiness lines may be better optimized to increase diversification anddecrease macroeconomic risks, or deliberately focused to exploitexpected advantages. Shareholders of companies also stand to benefitfrom an internal diversification measurement; such values provideinsight as to the focus, interrelationships and vulnerabilities of thecompany.

The only limitation towards the efficacy of investment portfoliodiversification stems from a belief in the relative attractiveness of aparticular asset. If other assets of comparable attractiveness cannot bediscovered, then there is a reasonable justification for holding lessdiverse portfolios. This constraint is really rather a constraint uponthe fund manager and the fund manger's resources including her own timefor studying, analyzing and predicting returns on investments. As aninvestment manager's resources grow, the manager's ability to discoverseveral highly attractive assets also increases and thus within aframework of rational participants and increasingly efficient marketsthe greater the importance of diversification in the investment policyand investment process. Without the ability to intelligently predictbasic portfolio optimization input assumptions, any investors would befoolish to deliberately accept a portfolio of less diversification.

DEFINITIONS

Assets: Assets are typically financial assets, such as stocks, bonds,funds, futures, derivatives, cash-based assets, real estate. However,assets may also be resources. The manager may uniquely define resources.A corporation, for example, may define a resource as a person,department, opportunity, intellectual property, process, product ornatural resource. Assets included the aforementioned financial assets,resources and statistical representations, derivatives, children orcomponent therefore.

Portfolios: Portfolios are collections of assets. Assets may vary inweight within a portfolio. Portfolios may consist of assets that haveallocation potential, but have no allocation. This scenario occurs whenconsidering an asset's allocation. Sometimes assets do not merit anallocation. However, the act of considering an asset provides it as amember of the portfolio. Portfolios may include assets of differenttypes. Portfolios may include collections of assets that vary over time.Portfolios may be either real or hypothetical. Portfolios must have atleast one asset, but are otherwise unlimited as to the number of assetsthey may contain.

Relationship: Normally, relationships of assets are given by statisticalmeasures. Correlation, co-variance, vector angles and cosines, singularor principal component values, semi-correlations, co-integration,copulas, R-squared and regression lines are all measures of assetrelationships. In addition to these known measurements of relationshipother approximations, measurements or estimations may be substituted.Substitutions would possess the property that any value will measure therelative relationship of each asset, or potentially, each asset to anexternal benchmark or index. Relationships may constitute otherquantifications of similarity, association, implication, proportion orrelativity. Such relationships may also depict an alternativeexpectation, such as a conditional measurement. In addition to theconventional measures of relationship above, for simplicity, thedefinition is extended to include conventional time series. Such timeseries values may depict a score, price, return value, simulated value,statistic or ratio. This includes time series and series of returninformation conventional in finance. Such measurement may vary over timeand probability and the measurements may be accounting for any valuethat is selected for which diversification will be measured.Relationship values may be mixed among assets. For example, some assetrelationships may be built with a historical correlation, while otherasset relationships in the same matrix may be built with a conditionedor estimated matrix. Relationships also include any derivative, subset,composite or estimation of any of the values obtained from theaforementioned methods.

Investor: This disclosure will also refer to an investor. An investorshall be taken to mean one who is allocating among assets, whether anindividual, institution, group, business process or softwareapplication. Investors may also include representatives, agents,employees, fiduciaries and advisors to the investors. Any user of asoftware application that is utilizing the processes described hereinwill also be construed as an investor.

Eigenasset: An eigenasset is the index value of the KL energy spectrum.Eigenassets may also be interpreted as the singular values of theassets.

Dimension: Because we are measuring diversification as a dimension, itis useful to distinguish several types of dimensions. This distinctionenables greater utility and precision.

Ambient Dimension: The ambient dimension for a portfolio of N assetswill be N. The ambient dimension is the initial vector space in whichthe data resides.

Intrinsic Dimension: The smallest number of parameters required to modelthe data without loss. Intrinsic dimension is equivalent to the spanningdimension. Within the context of our portfolio analysis, the intrinsicdimension is less than or equal to the ambient dimension. Thus as theintrinsic dimension is more narrowly defined it creates a more precisediversification measurement tool.

Spanning Dimension: The spanning dimension of the data is the standarddefinition for dimension in elementary linear algebra and indicates theminimum number of vectors required to span the data in the portfolio P.To formalize the definition consider the portfolio P consisting of |P|=massets. Let XP be the data matrix whose j-th column is the j-th vectorin the portfolio.

XP=[x1,x2 . . . xm−1,xm]  (1)

The spanning dimension is formally equivalent to matrix rank (XP).

Karhunen-Loève Dimension (KLD): The KLD reveals a latent or “natural”dimensionality of the portfolio when projected in a vector space. TheKLD is the dimension that is less than or equal to the intrinsicdimension. The KLD approximates the intrinsic dimension within aspecified degree of accuracy measured in terms of a confidence interval.The KLD is the spanning dimension of a subspace containing most of theoriginal portfolio data. The KLD approaches the intrinsic dimension, N,as the confidence interval (CI) approaches 0. Formally, KLD→N as CI→0

Karhunen-Loève (KL) expansion: KL expansion is used to reveal theapproximate intrinsic dimension of the portfolio. The KL expansion is ageneralization of a process also known by different names, depending onthe discipline applied. References to any one algorithm shall beinterpreted as applying any other equivalent algorithm. Other names forthe process are Principal Component Analysis (PCA), Hotelling Analysis,Empirical Component Analysis, Quasiharmonic Modes, Proper OrthogonalDecomposition (POD), Empirical Eigenfunction Decomposition and SingularValue Decomposition (SVD). Irrespective of the name of the function, itis a process used to construct an optimal basis for a subspace generallyused to reduce the dimensionality of a data set. While non-essential forthe description of the invention, for completeness, the followingtheorem as well as some results follows.

Theorem 1

Singular Value Decomposition (SVD) Let A be a real valued m×n matrix and1=min(m,n). Then there exist orthogonal matrices U and V such that

A=USV ^(T)  (2),

such that U is an element of R(m×m), V is an element of R(n×n) andS=diag(s1, . . . , sn) is an element of R(m×n).

With the data matrix defined in (1) we define the temporal covariancematrix as

Ct=X ^(T) X  (3)

We also define the spatial covariance matrix as:

C _(x) =XX ^(T)  (4)

Note that both matrices are symmetric and thus have a full set oforthonormal eigenvectors. The orthonormal eigenvectors are related tothe left and right singular vectors contained in U, and V respectivelyin the following way.

Proposition 1.1

The m left singular vectors of X exist and are given by the meigenvectors of Cx corresponding to nonzero eigenvalues. Theseeigenvalues correspond to the singular values squared.

Proposition 1.2

The n right singular vectors of X exist and are given by the neigenvectors of Ct corresponding to nonzero eigenvalues. Theseeigenvalues correspond to the singular values squared.

Proposition 1.3

Suppose C is an N×N symmetric matrix with zero mean then the eigenvaluesλi i=1, . . . , N are nonnegative.

Proposition 1.4

The eigenvectors of the spatial covariance matrix Cx are uncorrelated.

The above proposition is simply because the eigenvectors of a symmetricmatrix are orthogonal.

Proposition 1.5

For mean-subtracted data, the statistical variance of the j-thcoordinate direction is proportional to the j-th eigenvalue of C.

Proposition 1.6

Given a D term expansion in terms of the eigenbasis associated with C,the eigenvalues of C give a measurement of the truncation error:

$\begin{matrix}{{emse} = {\sum\limits_{j = {D + 1}}^{N}\; \lambda_{j}}} & (5)\end{matrix}$

Theorem 2

The basis defined by the spatial eigenvectors of Cx captures morestatistical variance than any other basis.

With this theoretical framework in place, we are now able to define theenergy of the dataset in terms of the statistical variance, orequivalently in terms of the singular values of the data.

The energy (EN) of the data set is defined as:

EN=sum(si), i=1, . . . , N  (6)

Thus, the energy captured by a D-term expansion (ED) is given by

ED=sum(si), i=1, . . . , D  (7)

We may use the normalized energy defined as

ED=ED/EN.  (8)

We will refer to a plot of the singular values (also Eigenassets) versusthe eigenvector index as a KL spectrum plot. It is often useful to plotsi/EN so that we can see immediately the fraction of the total energy(or variance) contained in each eigendirection. These plots are used toestimate the so-called KL dimension. This dimension is generally takenas the number of terms required to ensure that some minimum quantity ofenergy is captured by the data.

The KL energy dimension (simply KL dimension), written dim(KLEγ), isdefined to be the minimum number of terms Dγ required in the orthogonalexpansion to ensure that ED_(γ)≧γ

Confidence Interval: the confidence interval is used to associate theKLD with a probabilistic measure of certainty. The confidence intervalrelates to the diversification measure as such; there is a 95%confidence level that the portfolio diversification is XX.X. Confidenceintervals (CI) may be measured as intervals within a distribution ofpotential, real or hypothetical solutions. The KLD will approach thecount of the intrinsic dimension as the CI→1. This is the standardinterpretation of a confidence interval in statistics.

Conditioning Systems: Conditioning systems have the purpose and utilityto create better expected values and may be useful for prediction.Additionally, conditioning systems may also have the purpose to gainadditional diversification and performance insight under specialconditions.

Many conditioning systems are well known in the field of finance, themore standard econometric methods include Bayesian estimations,multi-factor regressions, moving averages, Markov chains, smoothing,GARCH models, Multi-sampling, neural networks, interpolations andextrapolations. Such techniques are applied at various points at theinvestor's discretion. For example, a conditioning system may be appliedto an input (time series, weight, relationship measure or confidenceinterval) or an output (visualization, metric, series of metrics or anentire database.)

KL diversification metric (KLDM): is defined as:

KLDM=Dγ/rank(X).  (13)

In other words, the KL diversification metric is the ratio of the KLenergy dimension and the intrinsic (or spanning) dimension of the data.Perfect diversification (from the feasible set) is achieved if the KLenergy dimension is equal to the intrinsic dimension of the data. TheKLDM thus presents diversification as a range of values from 0 to 1.Values approaching 1 has greater diversification. Higher values alsoshow that more assets inside the portfolio are contributing meaningfulmeasures of diversification.

SUMMARY

We are measuring portfolio diversification as the number of dimensionsthat the portfolio resides in. Portfolios with more dimensions have morediversification. Lesser-diversified portfolios are contained in fewerdimensions.

For example, unless otherwise conditioned in a zero correlation example(with all correlations=0), the portfolio dimensionality may be equal tothe number of assets in the portfolio (at least in the equally weightedscenario). As assets in a portfolio vary from zero correlation and tendtowards −1 or 1 the portfolio dimensionality will decrease. In anotherextreme example (and again existing in an unconditioned state), aportfolio consisting of all assets with a perfect 1 or −1 correlationwould have a dimensionality of 1.

To illustrate, consider FIG. 16, this may also be interpreted as havingeach asset be indistinguishable from one another. In this model, whenall assets share a perfect 1 correlation with one another each assets isindistinguishable from any other asset and from a portfolio perspective,any one asset adds no more diversification than an hypotheticalportfolio consisting of a singular asset. Just as a single point placedin a higher dimension, such as a point on a line the portfolio couldmove in only two directions. A point in 3 dimensions can move in moredirections that are independent. Each additional dimension explains anadditional level of independent performance. As such, a portfolioconsisting of a number of independent performing assets will have adimensionality equal to the number of assets.

Dimensions are inherently perpendicular or orthogonal to one another.For example, consider a two dimensional graph, the X and the Ydimensions are orthogonal (perpendicular) to one another. When thedimension is increased, it still holds that the third dimension isorthogonal to the other dimensions.

Dimensions greater than three lose much of the intuition, except for the4th dimension which is often considered time. Indeed, in this sense,time is independent, uncorrelated or orthogonal to the three physicaldimensions.

There is no mathematical limit to the number of dimensions and byrepresenting a portfolio geometrically, we can place this portfolio intoa vector space and calculate the proper dimensionality of the portfolio.The geometry of the portfolio is still valid, even if we lose ourgeometric intuition as the portfolio transcends into the abstract realmof higher dimensions.

In one preferred embodiment, measuring diversification comprises thefollowing steps that may be executed in various orders. Please see FIG.1 for a depiction of a possible interaction of the nine steps.

Step 1. Obtain or create a portfolio

Step 2. Apply any weighting to assets

Step 3. Obtain or create a relationship measure for the assets

Step 4. Obtain or create a confidence interval

Step 5. Apply any conditioning systems

Step 6. Model the data in a vector space

Step 7. Apply the Karhunen-Loève expansion process

Step 8. Calculate the Karhunen-Loève Dimension

Step 9. Publish the diversification value

Conventional applications of the dimensional calculation techniquesapplied as the core process are predominately concerned with dimensionalreduction. Such dimensional reduction techniques are standard inefficient image processing and decomposition, similar to speechrecognition, the KL expansion is used to efficiently reduce and simplifylarge datasets. Here the utility is for size reduction and efficientelectronic transmission. Dimension maximization is a rather novelconcept, even applied to the fields in which dimensional reduction iscommon.

Dimensional maximization, quantification, attribution, optimization andanalytics are unique to the field of management sciences andinvestments.

The dimensional quantification is also harmonic with a portfolio and itsrelative position to the outside world. For example, a portfolio of 15dimensions would be tantamount to a portfolio influenced by 15 exogenousand independent factors. Therefore, an increase in the number ofindependent factors that could influence the portfolio decreases thepotential for any one factor to cause significant harm. It is alsopossible to better understand and analyze a portfolio, even if theholdings are not known. For investment managers this is useful forreporting results to investors, regulators and risk managers withouthaving to disclose the unique holdings. The investor benefits fromhaving a holistic view of a portfolio and insights into diversificationand risk.

BRIEF DESCRIPTION OF THE DRAWINGS

While the specification concludes with claims which particularly pointout and distinctly claim the invention, it is believed the presentinvention will be better understood from the following description ofcertain examples taken in conjunction with the accompanying drawings, inwhich like reference numerals identify the same elements and in which:

FIG. 1 shows a flowchart describing an exemplary process executed on acomputer system;

FIG. 2 shows a time series for a portfolio consisting of 45 equallyweighted assets observed over 104 trading days;

FIG. 3 shows a KL spectrum plot for the portfolio of FIG. 2;

FIG. 4 shows a time series for the portfolio of FIG. 2 with the dominanteigenasset overlaid in dots;

FIG. 5 shows a time series for the portfolio of FIG. 2 with the secondmost energetic eigenasset overlaid in the dotted line;

FIG. 6 shows a time series for a portfolio consisting of 17 equallyweighted assets consisting of hedge fund indices observed over 181trading periods;

FIG. 7 shows a KL spectrum plot for the portfolio described in FIG. 6;

FIG. 8 shows a time series representing the resealed caricatures of thetime series shown in FIG. 6;

FIG. 9 shows a time series for the portfolio described in FIG. 6 withthe second most energetic eigenasset overlaid in the dots;

FIG. 10 shows the KL energy spectrum of the portfolio described in FIG.2 consisting of 45 assets with 90% invested in a single asset and theremaining 10% evenly distributed among the other 44 assets;

FIG. 11 shows the KL energy spectrum of the portfolio described in FIG.2 consisting of 45 assets with 90% invested in a single asset and theremaining 10% evenly distributed among the other 44 assets;

FIG. 12 shows a plot of the KL energy dimension as a function ofweighting a single asset with value ω and evenly dividing the remaining(1−ω) among all other assets for the portfolio described in FIG. 2;

FIG. 13 shows a KL energy spectrum computed for a weighted portfolio,ω=0.8 computed from time series data rather than correlations;

FIG. 14 shows a KL energy spectrum computed for a weighted portfoliowhere one asset has 80% of the investment allocated to it;

FIG. 15 shows an illustration of a one-dimensional portfolio;

FIG. 16 shows the KL energy plot of eigenassets at various confidencelevels;

FIG. 17 shows a flow diagram for an exemplary process relating toanalyzing the diversification of a portfolio;

FIG. 18 shows a block diagram depicting an exemplary system configuredto analyze the diversification of a portfolio; and

FIG. 19 shows a flow diagram for an exemplary process relating toanalyzing the diversification of a portfolio.

DETAILED DESCRIPTION OF THE INVENTION

The following description of certain examples should not be used tolimit the scope of the present invention. Other examples, features,aspects, embodiments, and advantages of the invention will becomeapparent to those skilled in the art from the following description. Aswill be realized, the invention is capable of other different andobvious aspects, all without departing from the invention. Accordingly,the drawings and descriptions should be regarded as illustrative innature and not restrictive.

The preferred embodiments of the invention can be implemented on one ormore computer(s) and or one or more computer networks, such as a localarea network (LAN), a wide area network (WAN), the Internet, personalcomputer or other device containing a sufficient processing resource.

In whole or in part, such a computer system contains variousembodiments, such as one or more server(s), client computer(s),application computer(s) and/or other computer(s) can be utilized toimplement embodiments of the invention. Illustrative computers caninclude, e.g.: a central processing unit; memory (e.g., RAM, etc.);digital data storage (e.g., hard drives, etc.); input/output ports, dataentry devices (e.g., key boards, etc.); etc. The invention is designedbut not limited to operation on a conventional computer system, howeverembodiments of the invention may be used on other processing devicessuch as servers, calculators, laptop computers and mobile devices suchas PDAs.

Client computers may contain, in some embodiments, browser or similarsoftware that can access the diversification metrics and embodiments. Inother embodiments, the values are inputted to other systems or storedfor archival or future retrieval.

In some preferred embodiments, the system utilizes relational databases,such as, e.g., employing a relational database management system tocreate, update and/or administer a relational database. Standard QueryLanguage (SQL) statements entered by an investor or called by anotherapplication or process may create, update retrieve or manipulatediversification metrics contained within the database or file as well asthe primary inputs to the diversification calculation.

Data in the system may originate in a flat file, database, random accessmemory, XML file or similarly formatted conduit. Time series orrelationship data often originates from an exogenous system such as afinancial data vendor or an exchange. Further, portfolio-weightinginformation can come from a custodian, brokerage or accounting system.Generally, conditioning systems and confidence intervals would bedetermined from an investor-designated process implemented with akeyboard, mouse or other machine human interface device, but it wouldalso be possible for these values to be inherited from another exogenoussystem.

Output and output embodiments may be published to the same or differentdata storage and retrieval system. Such a system would be accessed byapplications or other network and computer users. Publishing of themetrics may be directly displayed to a computer monitor or published toa database or webpage, thus configured as an input or display to aremote computer, document or display device.

An example of an embodiment of the invention executed on a computer isshown in FIG. 19.

Step 1. Obtain or Create a Portfolio

The invention is executed on a computer system that accepts a list orarray of assets comprising the portfolio.

The collection of assets for a portfolio may stem from other systemssuch as a portfolio accounting system, broker dealer inventories,outputs of portfolio optimization or asset allocation programs, resultsof sorts and filters, index components, or other system collectivelygenerating a portfolio.

Certain embodiments of this invention may be used recursively to producethe portfolio.

Step 2. Apply any Weighting to Assets

When an initial investment is not evenly distributed over all assets ina portfolio we say the portfolio is weighted. In this section, weintroduce weighted portfolios.

In order to extend the geometric approach to weighted portfolios we needto understand how weighting a portfolio affects the geometry of the setP. Recall that in an equally weighted portfolio we normalize all assetvectors to have unit length, thus the SVD algorithm extracts onlyorientation information. In simple terms, all assets reside on the unithyper sphere, the SVD extracts information about how the assets aredistributed onto the sphere. When a portfolio is weighted, more value isassigned to a subset of the entire portfolio. Thus, the intrinsicdimension of the portfolio should resemble the intrinsic dimension ofthe subset. To cast this observation into a geometric frame, it makessense to imagine that weighting an asset is equivalent to lengtheningthe vector an appropriate amount. Therefore, shifting the assets resultsin the centroid of the dataset moving in a direction favoring theweighted vectors and, in turn, the constrained optimization problemimplicitly solved by the SVD will rotate the orthonormal eigenbasis insuch a way as to minimize the projected residual of the weighted vectorand will thus lean more heavily in the direction of the weightedasset(s). We say a set of vectors is orthonormal to mean, the vectorsare mutually orthogonal (completely uncorrelated) and each vector hasunit length (on the unit hypersphere). For example, if P is a perfectlydiversified portfolio. Then after steps 1 and 2 of have been performed,the resulting set of vectors is orthonormal.

Equivalently, when the set of asset vectors in P are weighted thegeometry of P changes in such a way that the KL dimension is altered bygiving preference to particular directions in the ambient space.

W=[ω1 ω2 . . . ωm−1 ωm]  (14)

be a vector of weights for a portfolio of m assets where

Sum(ωi)=1, i=1, . . . , m  (15)

Then the weighted portfolio is represented with the weighted data matrix

Xω=[ω1×1 ω2×2 . . . ωm−1×m−1 . . . ωm×m].  (16)

Geometrically, we have stretched the vector lengths in proportion totheir concentration and contribution to the portfolio; however, we havenot changed the orientation of assets and therefore have retainedcorrelation information. The weighted portfolio has a different geometrythan the equally weighted portfolio and again the SVD may be employed inthe same way to probe the geometry of this set.

In the case of the weighted portfolio, whose data matrix Xw is definedin (16), the weighted temporal covariance matrix is defined as

Cω,t=(Xω)TXω.  (17)

Asset weights are obtained independent of the time series data, and thusa connection between Cω,t and Ct is given. The symmetric m×m weightsmatrix SWM is shown in (18).

$\begin{matrix}{{SWM} = \begin{pmatrix}\omega_{1}^{2} & {\omega_{1}\omega_{2}} & \ldots & {\omega_{1}\omega_{m}} \\{\omega_{2}\omega_{1}} & \omega_{2}^{2} & \ldots & {\omega_{2}\omega_{m}} \\\vdots & \vdots & \ddots & \vdots \\{\omega_{m}\omega_{1}} & {\omega_{m}\omega_{2}} & \ldots & \omega_{m}^{2}\end{pmatrix}} & (18)\end{matrix}$

Then,

Cω,t=Ct**SWM  (19)

where the ** operator is defined as component wise multiplication

As an alternative embodiment, step 4 may be replaced with anoptimization process in which the array of asset weights were variablesand the assets are moved within the vector space in a manner toengendering to minimize or maximize a fitness function, such asmaximizing the dimensionality.

Weights of assets summing to over 1 (100%) are said to be leveragedportfolios. The diversification metrics can be calculated on theleveraged portfolio or notional portfolios.

Portfolio weights are accepted into the system for sources such asinterface device like a keyboard or mouse, as well as obtained from anexternal system such as a portfolio accounting system, broker dealerinventories, outputs of portfolio optimization or asset allocationprograms, results of sorts and filters, index components, or othersystem collectively generating a portfolio.

Default assumptions may be made with portfolios having no pre-definedweighting scheme.

It is another embodiment of the invention to overlay another allocationschema. Such overlays are common in finance and may include a currencyoverlay, options overlay, dynamic hedging strategies, tax overlay,allocation and trade size rounding overlay, trader or manager overlay orother overlay technique traditionally practiced in the art. Multipleoverlays may be applied and overlays may be applied in conjunction withother systems. In the event that an overlay is applied, thediversification metrics may be published with and without the overlay toshow the affect of the overlay program.

It is appropriate to measure diversification in harmony with the samelevel of management discretion, e.g. a decision to hire a manager whohas ultimate authority of assets under management, that would beoperated as if all the mangers holdings were one single asset.

Diversification metrics are calculated by combining the overlay with theportfolio and portfolio assets. In one embodiment, the system accepts anarray of overlay weights corresponding to the portfolio assets andmultiplies the time series of the assets by that of the normalized ormean subtracted overlay time series, thus obtaining a new relationshipand weight matrix. In another embodiment, the overlay programdiversification is separately calculated.

A further embodiment of the invention may allocate subsets of theportfolios to entities wishing to hedge, purchase, leverage or obtainexposure to a particular dimension. In such a case, the assets of theportfolios or portions thereof may be allocated to a particulardimension that most closely relates to a particular variable sought bythe obtaining entity. Such an allocation is obtained by capturing thedesignated energy of an isolated dimension by the principal methods usedin calculating the KL energy spectrum.

Step 3. Obtain or Create a Relationship Measure for the Assets.

Correlation may be the preferred relationship measure; however, otherrelationship measurements may also be utilized.

The relationship measure is equivalent whether built from time seriesdata or correlations. The process for performing the calculationsstemming from time series input follows.

Each asset in a portfolio P is represented as a time series consistingof n days of trading data. Let xi,j be the j-th observation of asset iwhere i=1, . . . , |P| (|P| is the number of assets in the portfolio)and j=1, . . . , n. For notational convenience, we will drop thesuperscript j corresponding to observation unless it is necessary toreference a particular observation. The time series may contain indexvalues, percent change, or asset value. In any case, we embed each assetinto Rn by treating each asset time series as an n dimensional vector.Thus, an entire portfolio of assets can be regarded as a set of points(vectors) contained in Rn, i.e. P is a subset of Rn. In this frame, wemay bring to bear the full theoretical power of linear algebra. Inparticular, correlations amongst assets manifest themselves as welldefined geometric structure in the algebraic set P. The placement of theassets in said capacity provides an ability to study the underlyinggeometry induced by a portfolio in order to quantify features of aportfolio.

In light of our representation of a time series as a vector, it isnatural to define the norm of a time series of length $N$ using thestandard two-norm definition from elementary linear algebra, ∥x∥,denoted and defined as:

∥x∥2=sum[(xj)̂2], j=1, . . . , N.  (2)

The mean value of a time series xk consisting of N observations isdefined as:

<x>=1/N*sum(xk,j), j=1, . . . , N.  (3)

It is standard practice to omit the vector time series index k for termsinside the bracket. It is also customary to mean subtract the timeseries and to work with the so-called caricature of the time series.

DEFINITION

The quantity x′=xk−<x> is called the caricature, or fluctuating field ofthe time series xk.

Calculating relationship measures from correlations is a standardprocedure detailed as follows.

Computing correlation matrices (or relations) from time series data is aroutine procedure. To compute a correlation matrix from time seriesdata:

1. For each time series x_(j), corresponding to the jth asset inportfolio P, compute the caricature {tilde over (x)}_(j)=x_(j)−<x_(j)>.

2. Normalize each caricature {tilde over (x)}_(j)={tilde over(x)}_(j)/∥{tilde over (x)}_(j)∥ so that ∥{tilde over (x)}_(j)∥=1.Geometrically, the portfolio is now characterized by a set of points onthe unit hypersphere.

3. Form the data matrix X=[{tilde over (x)}₁ {tilde over (x)}₂ . . .{tilde over (x)}_(N−1) {tilde over (x)}_(N)] where N=|P| (number ofassets in portfolio P).

4. Compute the spatial covariance matrix X^(T)X. By construction, thisis now a correlation matrix since the (i, j)th element of the matrix is[X]_(i,j)={tilde over (x)}_(i) ^(T){tilde over (x)}_(j)/(∥{tilde over(x)}_(i)∥ ∥{tilde over (x)}_(j)∥). When the time series have zero mean,as the caricatures do, then [X]_(i,j) is the cross correlation (this isthe precise definition of correlation found in nonlinear time seriesanalysis and statistics) between time series i and j. Geometrically,this is the dot product of two unit vectors, which is equal to thecosine of the angle between the two vectors.

The size of each matrix may be a function of the number ofinvestor-selected assets to be modeled. When an investor has selected Nassets, each matrix has a size of N.times.N. The correlation matrix maybe symmetric. A symmetric matrix here means that the elements of theupper right are identical to the matrix elements in the lower left. Thediagonal separating the matrix is the correlation of each asset withrespect to itself, and in this embodiment may be equal to one (1). Therequired inputs of the model may only require unique relationships.There is no limit on the number of assets that may be modeled using thepresent invention. N has no upper limit. There is no lower limit, butany portfolio must be comprised of at least 1 asset, such having adimensionality of 1.

It is often the case that portfolio managers have access to correlationmatrices as opposed to time series data. Alternatively, they have accessto time series data; however, the data has asynchronous starting and enddates, may have missing values. Given this, it is necessary to make theconnection between the two. Recall proposition 1.2 that states that thesingular values of a data matrix X are equal to the square roots of theeigenvalues of the temporal covariance matrix Ct=XTX. In this frame, thetemporal covariance matrix is precisely the correlation matrix inquestion.

Step 4. Obtain or Create a Confidence Interval

Note that this definition may utilize making a selection of rank. Rankselection processes can be made by arbitrary methods, such as 90%, 95%or 99%, representing popular confidence intervals used in statistics andsome risk measurements. The rank selection can also be determined byexamining the derivates of the relationship measures. For example, for acorrelation of R, one or more data conditioning techniques such asresampling or taking rolling observations may be applied. From this dataarray, one embodiment calculates a standard deviation to ascertain thevariability of the inputs. A ranking measurement can be then maderelative to the underlying data variability. Rank selection may also bedetermined by portfolio objectives. For example, portfolios havingimposed stringent constraints, such as asset liability ratios, timehorizons or hurdle rates would have rank selection related to theimposing constraints.

FIG. 16 shows various confidence levels associated with a KL energyspectrum. The confidence interval of 90% corresponds to 16 dimensions.CI=95% corresponds to 22 dimensions. CI=99% corresponds to 31Dimensions. Finally CI=99.5% corresponds to 36 dimensions.

Confidence intervals can be determined by investor input using machineinterface devices such as keyboards and pointing device. CI may also begiven by another system, or omitted for certain embodiments notrequiring the single KLD value.

In addition to inputting confidence intervals determined by the solutionquality of other optimizations, confidence intervals may also bemeasured versus benchmarks. Benchmarks may be reflecting actual orhypothetical performance of known portfolios. The tracking error of aportfolio juxtapose its benchmark may qualify as a given confidenceinterval. Managers with an edict to track a benchmark within a specifiedlevel of tracking error could infer a confidence interval relative tothe level of tracking error.

The dimensionality of the original matrix can be reduced using anyoptimization technique including a genetic optimization such as theTechnique described in Damschroder U.S. Pat. No. 7,472,084 theoptimization reduces the dimensionality while simultaneouslyaccumulating an error allowance that is the subject of a errorminimizing fitness function.

The total error value reflects the difference between the sum of errorsor squared errors of the relationship matrix and the vector cosinematrix.

The optimization engenders to minimize the fitness function that isequal to the sum of all errors. This is an error minimization functions.

The magnitude of the error expressed as a fraction of the totalpotential error gives a fraction that becomes the basis for a confidenceinterval. This fraction could first be amended to reflect populationsizes, derivatives of the matrix of its values, or results of externalor ancillary equations.

Step 5. Apply any conditioning systems

The core process can be subject to a variety of conditioning systems.

Data conditioning may take place at one discrete step, or may actuallyoccur at one or more steps in the system. At any point that anintermediate value is obtained or created such a values or series ofvalues may be affected by applying a conditioning system.

Conditioning systems typically represent stored routines executed on acomputer processor that apply to measurement inputs and outputs.

In addition to calculating relationship measures on the asset directly,sometimes it can be advantageous to measure diversification only as itpertain to one element of analysis (e.g., Factor.) For this purpose, therelationship measure(s) can be applied to elements in common forportfolio assets. For example, to analyze diversification solely as itpertains to international exposure one can isolate these elements andanalyze only those parts. Traditional methods can be used to isolatesuch elements, such as principal component analysis, allocation weightsor attribution exposures for common statistics such as risk or return.

This method may be applied to either the total exposure to an element ofanalysis; such as one country among an array of potential countries orit can be applied per asset by combining the asset with asset weightsand the elements and element weights.

Correlations or other relationship measures may be normalized to accountfor negative correlations. A negative correlation takes the valuesbetween zero and −1. Such relationships move at least partially inopposition to one another. In the core process, large negative valuesare considered deterministic and would adversely affect the totalportfolio diversification, at least relative to a non-correlated asset.However, for investment managers the impact of negative correlationvalues on total diversification may be subjective.

To assume that negatively correlated assets increase diversification amanager would be assuming, or otherwise applying the idea that therewould be some uncoupling between the assets expected prices, otherwisepositive returns from one asset would be negated by an equal negativemovement from it's counterpart. This is an assumption that some managersfeel comfortable making while another manager may not. To enable theinvestor who is comfortable predicting the divergence of the historicalvalues, they may be able to achieve greater diversification by usingnegatively correlated assets.

A correlation value may be conditioned to reflect this disposition withthe following operation:

=(1+Correlation)/2

The result of which could be further conditioned so that the resultwould be fairly comparable to unconditioned values. This could becalculated by a normalization function.

This embodiment has an additional characteristic in that the assetshaving a significant, but negative relationship with one another are maysummarily be characterized as a zero correlation.

Such techniques can also be a conditional filter, based on themeasurement, trend, value, or derivative of one or more economicvariables. For example, it could be desirable to measurementdiversification for a portfolio during periods of historically highinflation rates. In such case, relationship data could be filtered tothose periods of history specifically matching the filter.

Another conditioning system would involve combining relationshipmeasures of varying sampling frequencies. The weightings of suchcalculations can be based on the uncertainty of the future horizon forwhich we are creating the analysis or measurement. Consider thecircumstance where an investor's intended holding period was one year,plus or minus three months. We can combine one sample of relationshipmeasures based on a price time series with a second sample based on areturn series. The return series and the weight of the sample would beproportional to the certainty of the time horizon. In this example, theinvestor would have a minimum time horizon of nine months and a maximumtime horizon of 15 months. This range of six months is expressed as asample weight relative to the expected time horizon. Therefore, in thisexample the investor would weigh the return-based series 50% of theprice-based series. The result provides a relationship measure that isrelative to the future and the future uncertainty.

Diversification metrics may be built on a single period, multi-period,or amalgamation of such periods. Periods may correspond to othervariables besides time, such as probability, or similar sort order.

The measurements of diversification may be affected by a variety ofstatistical processes which may occur at various times in theconstruction of the metrics. Simulation, re-sampling, extrapolation:each may occur with the asset time series, relationship measurement,dimensional reduction process and confidence interval determination.Alternatively, simulations may be applied ex post to the results of thedimensional measurement and publishing process.

Step 6. Model the Data in a Vector Space

To model the data as required for calculations, all the input data iscollected and the data can be interpreted as being placed in a vectorspace. Implementations of this step are often automatic provided thatthe dimension calculation algorithms executed on computer processorsinterpret the data in a geometric fashion. Other algorithms may requirethe asset weightings to be combined with the relationship measures toenable the geometric interpretation.

a. Compute the symmetric weights matrix SWM.

b. Compute the weighted temporal relationship matrix Cω,t.

c. Compute the eigenvalues λi of Cω,t

Step 7. Apply the KL Expansion Process

From the following steps, we have assembled the necessary inputs tocompute the KL energy spectrum.

d. Compute sqrt(λi) for every i, these are the singular values of theunknown data matrix Xω.

In the preferred embodiment, the inputs are obtained from a computerrandom access memory and a computer processor executes a routinecontaining the Karhunen-Loève or equivalent algorithm as described inthe definition for Karhunen-Loève (KL) expansion.

This process creates an output containing a series of eigenassets (alsoknow as the singular values) This output is stored in a computerreadable media and passed to step 8.

Step 8. Calculate the Karhunen-Loève Dimension

The KL dimension is then given by the associated rank of the assetscorresponding to the inputted confidence interval. If the spectrum iscontinuous, then the KL dimension calculation is provided by integratingthe spectrum up to the confidence interval.

The result of the KL expansion are associated with the confidenceinterval. In one embodiment, a calculation is executed on a computerprocessor that accepts the output of the KL energy spectrum as well asaccepts the CI. The algorithm simply counts the eigenassets ordered fromgreatest to least and returns the value most closely associated with theconfidence interval. Other variations include returning the last countedvalue prior to reaching the sum determined by the CI.

As an example, in a 100 asset portfolio, the sum of the first 65eigenassets equals 94.5% of the total energy. If the inclusion of the66^(th) ranked eigenasset would cause the total energy to sum to 95.3,the algorithm would stop at 65 and return a KLD=65 provided a CI=95%.

The KL energy spectrum can be interpreted as either a discrete spectrumor continuous spectrum. In the discrete case, the KLD is computed bysummation, and in the continuous case, the KLD is computed byintegration. This enables fractional KLD measurements. Fractional KLDmetrics are especially useful in low dimension portfolios.

To transform a discrete KL spectrum one would only need to apply aninterpolation scheme to the spectrum.

As an alternative embodiment, steps 7 and 8 can be replaced by the twosymmetry quantification methods described below.

A second method is provided for the diversification measurement processwherein the amount of diversification is a function of the degree inwhich a portfolio is symmetrical. The portfolio is projected into avector space by mapping vectors cosines to correlations. The shape ofthe portfolio can be determined by methods disclosed in U.S. Pat. No.7,472,084, which is incorporated herein by reference.

Another embodiment places the assets in a vector space. Vectors aremapped by equating the direction from the origin. Correlations of assetsare mapped to cosines of the vectors. Vector lengths are given by aninvestor defined utility function.

With the polytope determined, one can understand diversification bymeasuring the shape of the portfolio to a model of perfect symmetry.This measurement is known as sphericty. Sphericty measures the volume tosurface area ratio of a polyhedron or polytope.

The computation may be taken as:

The hyperdimensional volume of the space which a (n−1)-sphere encloses(the n-ball) is given by

$V_{n} = {\frac{\pi^{\frac{n}{2}}R^{n}}{\Gamma \left( {\frac{n}{2} + 1} \right)} = {C_{n}R^{n}}}$

where Γ is the gamma function. (For even n,

${\Gamma \left( {\frac{n}{2} + 1} \right)} = {\left( \frac{n}{2} \right)!}$

for odd n,

${{\Gamma \left( {\frac{n}{2} + 1} \right)} = {\sqrt{\pi}\frac{n!!}{2^{{({n + 1})}/2}}}},$

where n!! denotes the double factorial.

From this, it follows that the value of the constant Cn for a given nis:

${C_{n} = \frac{\pi^{r}}{r!}},$

for even n such that n=2r, and

${C_{n} = \frac{2^{{({n + 1})}/2}\pi^{{({n - 1})}/2}}{n!!}},$

for odd n.

The “surface area” of this (n−1)-sphere is

$S_{n} = {\frac{V_{n}}{R} = {\frac{{nV}_{n}}{R} = {\frac{2\pi^{\frac{n}{2}}R^{n - 1}}{\Gamma \left( \frac{n}{2} \right)} = {{nC}_{n}R^{n - 1}}}}}$

The following relationships hold between the n-spherical surface areaand volume:

V _(n) /S _(n) =R/n

S _(n+2) /V _(n)=2πR

As an alternative to the sphericty as a measure of symmetry, othermeasurement of symmetry may be substituted. FIG. 19 shows an embodimentof the invention using symmetry quantification.

A further embodiment of the invention can be produced by dividing theKLD obtained in step 8 by the intrinsic dimension of that portfolio. Theresult produces a ratio explaining the diversification level for a givenset of assets, relative to the total diversification potential for thoseassets. This ratio would be equal to one if the KLD and the Intrinsicdimension would be equal. This value would tend towards zero as theportfolio became relatively less diversified. This is similar to the IPCmetric, but effectively perfect diversification when all assets areuncorrelated, whereas the IPC interprets perfect diversification whenall assets are negatively correlated.

Step 9. Publish the Diversification Value

The output of the diversification quantification process is thenpublished to another media, such as a printer, computer display,database, external system. The output may also be combined with othervalues and published.

This measurement may then be conjoined with other measurements in eithera time-weighted or capitalization-weighted manner. The result of whichwould produce a diversification measure transcending time and changesmade the composition, relationships and weightings of the portfolio.

Following the method in A, it can be then augmented to exhibit thepassing of time. Here time can be modeled continuously or at anyinterval given as an input to the system or derived from otherinformation. With each passing of time providing a new location of eachasset and its diversification.

Another output of the system can be configured to produce a chart whichdisplays a line or area graph of the diversification metric over time.This can be applied to the same portfolio with the same weights or withvarying weights.

Another output of the system can be configured to produce a chart whichdisplays a probability distribution the diversification metric. Such adistribution may be produced using simulation or sampling techniques orhistorically observed.

Another output of the system can be configured to produce a chart whichdisplays a 3D contour surface of the data displaying the diversificationmetric probability or confidence interval of the measure, and time.

Another output of the system can be configured to produce a chart whichdisplays a graph of the diversification metric and the probabilitylevel.

Another output of the system can be configured to produce a tabularmatrix comprised of the diversification metrics and confidenceintervals.

Another output of the system can be configured to produce a chart whichdisplays a comparative diversification metric of multiple portfolios asa radar graph.

Another output of the system can be configured to produce a chart whichdisplays comparative diversification metric of multiple portfolios overtime as line graphs.

Another output of the system can be configured to produce a chart whichdisplays comparative diversification metrics of portfolio components asa pie chart.

Another output of the system can be configured to produce a chart whichdisplays comparative diversification metrics of multiple portfolios as abar graph.

Another output of the system can be configured to produce a chart whichdisplays comparative diversification metrics of multiple portfolios aswell as portfolio risk and return data as a 3D contour or surface chart.

Another output of the system can be configured to produce a chart whichdisplays comparative diversification metrics including the KLD, KLDM,IPC, and CC.

Another output of the system can be configured to produce a chart whichdisplays the relative diversification contribution of each portfolioasset.

Another output of the system can be configured to produce a chart whichdisplays comparative diversification metrics of multiple portfolios, aswell as assets' other fundamental metrics, such as risk, return, fees,performance ratios, valuation metrics or other forms of quantitativeanalysis.

Another output of the system can be configured to produce a chart whichdisplays comparative diversification metrics of multiple portfolioswhere the diversification metric determines the size of the displaypoint. The display point may be a circle, sphere, X or other object. Thelocation of the point may be given by other metrics.

Another output of the system can be configured to produce a chart whichdisplays the relative ranges of diversification corresponding to variousconfidence intervals and shows how these ranges fluctuate over time.

Another output of the system can be configured to produce a chart whichdisplays the relative diversification metrics against a population ofother portfolios and their diversification metrics.

Example outputs are shown in FIGS. 2-16. FIG. 2 depicts an output for anexample portfolio consisting of 45 equally weighted assets observed over104 trading days. The time series represents the resealed caricatures ofthe original time series. A band of highly correlated behavior is existsas evidenced from the output.

FIG. 3 depicts a KL spectrum plot for the portfolio depicted in FIG. 2consisting of 45 equally weighted assets observed over 104 trading days.The plot is a normalized energy plot. The largest value (far left) isabout 0.28 which implies 28% of the portfolio's variance is described inthe dominant eigenasset direction, 95% of the variance is captured by 27eigenassets and the rank of the portfolio is 44 (number of nonzerosingular values). FIG. 4 depicts the total portfolio from FIG. 2 withthe dominant eigenasset overlaid in dots. Approximately 28% of theportfolio is explained by the performance of the dotted trend line.

FIG. 5 depicts the asset allocation portfolio shown in FIG. 4 with thesecond most energetic eigenasset overlaid in the dotted line.Approximately 9% of the portfolio's variance is captured by the dottedtrend line. Adding eigenassets' together until the sum of the variancesmeets the confidence interval is a way to determine the dimension.

To contrast the previous example, consider a new example having fewerassets. FIG. 6 depicts an asset allocation portfolio consisting of 17equally weighted assets consisting of hedge fund indices observed over181 trading periods. The time series represents the resealed caricaturesof the original time series. This portfolio is less diversified than theexample in FIG. 2, in the sense that the KLD is lower. However, theamount of diversification relative to the portfolio's potential isgreater, thus it has a greater KLDM.

FIG. 7 depicts the KL spectrum plot for the hedge fund indices portfoliorelating to FIG. 6 that consists of 17 equally weighted assets observedover 181 trading days. This is a normalized energy plot. The largestvalue is about 0.23 which implies 23% of the portfolio's variance isdescribed in the dominant eigenasset direction, 95% of the variance iscaptured by 14 eigenassets and the rank of the portfolio is 16 (numberof nonzero singular values).

FIG. 8 depicts the portfolio relating to FIG. 6 that consists of 17equally weighted assets consisting of hedge fund indices observed over181 trading days. The time series represents the resealed caricatures ofthe original time series. The dominant eigenasset is overlaid in dotsand contains 23% of the portfolio's variance. Whereas FIG. 9 depicts thehedge fund indices portfolio with the second most energetic eigenassetoverlaid in the dots. Approximately 12% of the portfolio's variance iscaptured by the dotted trend line, each dot representing the periodiceigenasset.

As shown in FIG. 10, the first eigenasset contains 90% of theportfolio's resources and the remaining 10% is evenly distributed to theremaining 44 assets. This is not a diversified portfolio. This portfoliois almost 1 dimensional and the KL energy dimension confirms thisintuition. FIG. 10 shows the KL energy spectrum of the weightedportfolio. A portfolio consisting of 45 assets with 90% invested in asingle asset and the remaining 10% evenly distributed among the other 44assets. The dominant eigenasset overlaid in the dotted line containsnearly 98% of the portfolio's variance. It is visually apparent that thedominant eigendirection is nearly identical to the highly weighted assetas expected.

FIG. 11 depicts a KL energy spectrum of a portfolio consisting of 45assets with 90% invested in a single asset and the remaining 10% evenlydistributed among the other 44 assets. The heavy weighting in onedirection is apparent and severely reduces the KL energy dimension ofthe portfolio. This portfolio is 1 dimensional at the 95% confidenceinterval since the dominant eigenasset contains over 95% of thevariance. The dominant eigenasset is shown on the far left of FIG. 11.

FIG. 12 depicts a plot of the KL energy dimension as a function ofweighting a single asset with value ω and evenly dividing the remaining(1−ω) among all other assets. Computing the KL energy dimension isachieved by taking the portfolio in FIG. 2 and weighting a selectedasset by ωi and then evenly allocating the remaining (1−ωi) amongst theremaining assets in the portfolio. In FIG. 12, ω=0 represents the KLdimension of an equally weighted portfolio. The initial jump shown inFIG. 12 shows where diversification is maximized for the weighting ofthe principal asset. For the portfolio studied here, the equallyweighted portfolio has a KL energy dimension of 27. The KL energydimension decays at a nearly linear rate with respect to weight.

FIG. 13 depicts a KL energy spectrum computed for a weighted portfolio,ω=0.8 computed from time series data rather than correlations. Theremaining 20% is evenly distributed among the other assets. FIG. 14depicts a KL energy spectrum computed for a weighted portfolio, oneasset has 80% of the investment allocated to it. The same assets andsame weights are used as applied in FIG. 13 but this portfolio is builtfrom correlations. The results are identical as those obtained directlyfrom time series observations.

While FIG. 15 depicts an illustration of a one-dimensional portfolio,FIG. 16 shows the KL energy plot of the eigenassets. Confidenceintervals of 90, 95, 99 and 99.5% have been depicted.

FIG. 17 shows a process flow diagram of the invention executed on asystem such as that shown in FIG. 18 wherein system (10) includes aninput device (20) in communication with a processor (30). Processor (30)is operatively connected with a computer readable medium (35).

The term “computer readable medium” should be read broadly to includeany object, substance, or combination of objects or substances, capableof storing data or instructions in a form in which they can be retrievedand/or processed by a device. A computer readable medium should not belimited to any particular type or organization, and should be understoodto include distributed and decentralized systems however they arephysically or logically disposed, as well as storage objects of systemswhich are located in a defined and/or circumscribed physical and/orlogical space.

When used in the claims, “operatively connected” may be understood torefer to a relationship where a thing is able to control or influencethe operation of the thing to which it is connected. Examples toillustrate the meaning of “operatively connected” include a processoroperatively connected to a computer-readable medium (where the mediumstores instructions which can control the tasks performed by theprocessor), a printer operatively connected to a server (for example,where a server communicates over a network with a printer to indicatewhat should be printed), and a monitor operatively connected to aprogram (where the program might control, perhaps in combination with amedium and a processor, what is displayed on the monitor).

As shown in FIG. 17, an investor may input asset data to create aportfolio. This may be achieved in a variety of ways including forexample using an input device (20) having a user interface. An exemplaryuser interface may be understood to refer to one or more tools whichallows a user to interact with an automated system. However, anysuitable user interface may be used that allows a user to input assetsinto a portfolio.

Several events occur after the portfolio is in existence including theretrieval of a time series from a database or storage device.

Correlations or relationship measures are then also computed using aprocessor (30). Any suitable technique may be used for computingcorrelations and relationship measures including those described earlierin “Step 3. Obtain or create a relationship measure for the assets.” Theverb “retrieve” (and the various forms thereof) when used in the contextof data should be understood to mean reading the data “retrieved” from alocation in which that data is stored, including for example a database(40). Database may be understood to refer to an organized collection ofdata. Similarly, a “data source” may be understood to refer to acomputer-readable data structure for storing a collection of dataincluding, but not limited to, databases, data warehouses and datamarts.

Likewise, any suitable processor may be used. For example, a processormay include a component integrated into a computer, which performscalculations, logical operations, and other manipulations of data. Acomputer may be understood to refer to a device or group of deviceswhich is capable of performing one or more logical and/or physicaloperations on data to produce a result. Computer executable instructionsmay be understood to refer to data, whether stored as hardware,software, firmware, or some other manner, which can be used to specifyphysical or logical operations to be performed by a computer orprocessor.

Also occurring after a portfolio is in existence, asset weightings maybe inputted using for example an input device (20) that may or may notbe the same as the device used to create the portfolio. Any suitabledevice or technique may be used to input data into the system. The inputof data may be automatic or manual including the input of assetweightings. The term “data” should be understood to mean informationwhich is represented in a form which is capable of being processed,stored and/or transmitted. Hence, data would include but not be limitedto asset information, time series information, asset weightings, dataconditioning process information, and so on. Any suitable technique maybe used for weighing assets including those described earlier under“Step 2. Apply any weighing to assets.” Likewise, any suitable techniquemay be used for applying any condition systems including those describedearlier in “Step 5. Apply any condition systems.”

In the example shown in FIG. 17, all of the input data is assembled forprocessing wherein processing includes modeling the data as required forcalculations. The data may be modeled in vector space using any suitabletechnique including those described earlier in “Step 6. Model the datain vector space.” For this example, input data would include but not belimited to correlations, relationship measures, asset data, assetweightings, data condition process information.

After the data is assembled, the KL expansion routine is applied asexecuted by processor (30). Processor (30) may also accept or determinea confidence level based on the investors input or a database (40). Anysuitable technique may be used to apply the KL expansion including thosedescribed earlier in “Step 7. Apply the KL expansion process.” Anysuitable technique may be used to determine a confidence level includingthose described earlier in “Step 4. Obtain or create a confidenceinterval.”

After the KL expansion is applied, processor (30) calculates the KLdimension. Any suitable technique may be used to calculate the KLdimension including those described earlier in “Step 8. Calculate theKarhunen-Loève Dimension.” The output of the diversificationquantification process is then sent to a database (40). Any suitableoutput may be sent including a chart, tabular matrix including thosedescribed earlier in “Step 9. Publish the diversification value.”Visualizations, reports, etc. may then be displayed including forexample on the user interface of an input device (20). Thevisualizations, reports, etc. may be displayed on an output device (50).The output device may be understood to mean a device which presents datato a user. Examples of output devices include monitors (which presentdata in a visual form), and speakers (which present data in auditoryform). Any suitable output device may be used that displays data to auser. The output data will remain stored in a database (40) for use byexternal systems.

FIG. 19 shows a process flow diagram of the invention using symmetryquantification as an alternative embodiment. The numbers associated withthe flow diagram are not necessarily indicative of any particular orderin which the process occurs. As shown, a determination of a portfolio ismade where the portfolio includes more than one asset. Any suitablesystem, technique, or structure may be used to make this determination.For example, a user may utilize a user interface through an input device(20) to create a portfolio. An automated system may applied that createsa portfolio. After the portfolio is in existence, asset weightings maybe applied. Likewise, relationship measures may be created or obtained.Similar to the creation of the portfolio, the application of assetweightings and relationship measures may be automatic or manual, or acombination of both. With respect to the relationship measures and assetweightings, data condition processes may be applied to each as well.

The assets may then be modeled in vector space by a processor (30) afterapplying the asset weightings and relationship measures. Further, apolytope representation of the portfolio may be created based in part onaccepting or determining a confidence level. Once the representation iscomplete, the portfolio symmetry is calculated. The diversificationmetrics and other related outputs are then made available based on thecalculated portfolio symmetry. The output data may be displayed on anoutput device (50).

A further embodiment of the inventions calculates a compositediversification metric. Such a diversification metric may either treat aportfolio of assets as a single asset, or reach inside and create acomposite of the individual assets depending on the investor selectionor business process.

It is an embodiment of the invention to perform a diversificationsearch. In this embodiment, steps 1 and or 2 are iterated with changesmade to the assets or weights. A diversification search, may represent aprocess wherein an investor seeking to add diversification to aportfolio may query a universe of investment candidates and rank theresult in order to learn which assets provide incrementally morediversification to a portfolio. A diversification search is thereforeuseful to help investor change and improve a portfolio. The search maybe performed by calculating the KL dimension for the portfolio anditeratively add assets, recalculate the KL dimension. Assets are thenarranged by how they affect the overall dimension. This process can bereadily adapted to account for asset weighted by multiplying the assetby the weight in the manner described in step 2. The process may also beadapted by multiplying the assets by a utility function. Standardoptimization processes such as gradient searches, linear programming andevolutionary search can replace the iterative cycle to improveperformance.

If we use a utility function instead of a weight array, then compute,then we re-compute without the utility function the difference in theSVD could explain an optimal step

It is further an embodiment of the invention that an investor coulddesignate a certain quantity of assets to invest in. Otherwise, such aportfolio constraint may be arrived at with an exogenous system. Withsuch a criteria in place, the iterative search feature described in theprior paragraph may also be looped until a constraint is reached.Additionally if the investor selected a preference for more or lessquantity of holdings then they could arrive at the diversification levelby summing the energy levels of the assets from largest to smallest orsmallest allocation weight to largest weight until the designateddiversification measurement was ascertained.

An alternative method to publish the results would display thediversification metrics relative to the individual assets, rather thanthe portfolio.

Once a portfolio has been processed and diversification metricsacquired, it is possible to attribute the amount of portfoliodiversification to the various assets, factors, periods or statisticalcategories. The invention can be iterated with each asset removed or theweight set to zero, in such a process the new KLD measurements can berelated to the original KLD value and the difference attributed to theasset that was changed and then the difference may be divided the totalportfolio dimensionality. This fraction helps investors understand howany asset may affect diversification.

Most rudimentary is a process to determine the amount of diversificationgiven to an asset. For example, given a 9 dimensional portfoliocomprised of 20 equally weighted assets, we know that asset G which has5% of the total allocation weight. Using the attribution process, aninvestor could learn that asset G adds 1 dimension to the portfoliogiving it an attribution of 1/9=11.11%.

Having shown and described various embodiments of the present invention,further adaptations of the methods and systems described herein may beaccomplished by appropriate modifications by one of ordinary skill inthe art without departing from the scope of the present invention.Several of such potential modifications have been mentioned, and otherswill be apparent to those skilled in the art. For instance, theexamples, embodiments, ratios, steps, and the like discussed above areillustrative and are not required. Accordingly, the scope of the presentinvention should be considered in terms of the following claims and isunderstood not to be limited to the details of structure and operationshown and described in the specification and drawings.

1. A machine comprising: (a) a means for calculating a diversificationvalue for a portfolio based on at least one dimension derived from aKarhunen-Loève expansion for the portfolio; and (b) a means forpublishing the diversification value.
 2. The machine of claim 1 whereinthe means for calculating the diversification value for the portfolio isconfigured to: (a) receive the portfolio as input, wherein the portfoliocomprises a plurality of assets; (b) obtain a set of weight data byweighting each of the plurality of assets within the portfolio accordingto an investment value allocated to each of said assets; (c) obtainingrelationships between each of the plurality of assets and the set ofweight data; and (d) model the portfolio in a geometric space based onasset relationship and the set of weight data.
 3. The machine of claim 1wherein the means for calculating the diversification value for theportfolio is configured to: (a) receive the portfolio as input, whereinthe portfolio comprises a plurality of assets; (b) weight each of theplurality of assets within the portfolio according to an investmentvalue allocated to each of said assets; (c) obtain relationships betweeneach asset from the plurality of assets and at least one other assetfrom the plurality of assets by performing an act taken from the set ofacts comprising: calculating the relationships between each asset fromthe plurality of assets and at least one other asset; and, receiving therelationships between each asset from the plurality of assets and atleast one other asset; (d) model the portfolio in a geometric spacebased on asset relationship and weight data.
 4. The machine of claim 1,wherein the means for calculating the diversification value for theportfolio is configured to create a rolling time series of the at leastone dimension.
 5. The machine of claim 1, wherein the means forpublishing the diversification value is configured to create a chartbased on combining the diversification value with a set of portfoliostatistical data.
 6. The machine of claim 1, wherein the means forcalculating the diversification value for the portfolio is configured tocreate ratios of dimensions at one or more confidence intervals.
 7. Themachine of claim 1, wherein the means for calculating thediversification value for the portfolio is configured to determinerelative contributions of each asset in the portfolio to thediversification value for the portfolio, based on measuring theportfolio dimensionality with and without each asset.
 8. The machine ofclaim 1, wherein the means for calculating the diversification value forthe portfolio is configured to create a ratio of a dimension at aselected confidence interval to the portfolios' ambient dimension.
 9. Asystem for measuring the diversification of a portfolio, the systemcomprising a computer processor and a set of computer-executableinstructions configuring the system to perform a set of steps, the setof steps comprising: i. receiving, as input, a portfolio comprising aplurality of assets, asset relationship and weight data; ii. modelingthe portfolio in a geometric space using the asset relationships andweight data; iii. computing one or more dimensions for the portfoliobased on a Karhunen Loeve expansion for the portfolio; and iv. producinga diversification metric using the one or more dimensions.
 10. Thesystem of claim 9 wherein the set of steps further comprises: (a)creating a rolling time series of the one or more dimensions; and (b)using the rolling time series to detect changes in one or more elementstaken from the set consisting of: i. diversification; ii. systematicrisk; and iii. idiosyncratic risk.
 11. The system of claim 9, whereinthe set of steps further comprises: (a) combining the diversificationmetric with portfolio statistical data; and (b) creating a 3d contoursurface, with the 3 dimensions depicting values for portfolio risk,portfolio return and the diversification metric.
 12. The system of claim9, wherein the set of steps further comprises creating a ratio ofdimensions taken from the Karhunen-Loève expansion.
 13. The system ofclaim 9, wherein the set of steps further comprises determining relativecontributions of each asset in the portfolio to the diversificationmetric, based on measuring the one or more dimensions with and withouteach asset.
 14. The system of claim 9, wherein the set of steps furthercomprises comparing the one or more dimensions for the portfolio versusportfolio dimension data taken from the set consisting of: (a) a secondportfolio; (b) a sample of other portfolio diversification metrics; (c)a population of other portfolio diversification metrics; and (d) avariation of a conditioning system applied to the portfolio.
 15. Thesystem of claim 9, wherein the set of steps further comprises theapplication of a conditioning system to input data.
 16. The system ofclaim 9, wherein the set of steps further comprises creating one or moreratios of dimensions, wherein a ratio is determined from the ratio setand is taken as the diversification metric.
 17. A holistic portfoliodiversification analysis system which comprises a computer processor andis operable to produce an analytical framework for evaluating integratedrisks from both systemic and non-systemic sources, the system configuredaccording to a set of computer executable instructions encoded on acomputer readable medium to perform a set of steps comprising: i.receiving, as input, a portfolio comprising a plurality of assets; ii.obtaining a set of asset relationship and weight data for the portfolio;iii. modeling the portfolio in a geometric space using the assetrelationship and weight data; and iv. computing an array of dimensionsfor the portfolio based on a Karhunen-Loève expansion for the portfolio.18. The system of claim 17 wherein the set of steps further comprises:(a) creating a rolling time series of the array of dimensions; and (b)using the rolling time series of the array of dimensions to detectchanges in diversification.
 19. The system of claim 17, wherein the setof steps further comprises: (a) creating a rolling time series of thearray of dimensions; and (b) using the rolling time series of the arrayof dimensions to detect changes in systematic risk.
 20. The system ofclaim 17, wherein the set of steps further comprises creating ratios ofdimensions.
 21. The system of claim 17, wherein the set of steps furthercomprises determining relative contributions of each asset in theportfolio to a diversification value for the portfolio, based onmeasuring the array of dimensions with and without each asset.